280 
MR E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
finite value, can be made as small as we please by taking | x | sufficiently large. The 
integral is taken in the positive direction round a small circle enclosing s = -9. 
The integral is equal to 
P°s (~ x )t K„ 
v 7 n\ 
71 = 0 
where K„ is the coefficient of e (n+1> in the expansion of T (0-e) exp {l/e}. 
We therefore have 
* (-)»r ™(Q) 
n ?u=o m ! (m+n+ 1)! 
Hence, when i\ (x) < 0, 
f(x \ = ( _ x) ~, i [%(-*)]'• s (-ruuq +J(x) , 
where | J (x)x l \ for any finite value of l tends to zero as \x\ tends to infinity. 
The double series obviously represents an integral function of log (-as), and may 
be written 
y " 
| | __ 
Hi -o 77i ! n=o n ! (m +1 + n )! 
, where y = log ( —x). 
Using the notation oFfp ; x } for the series 
y n r ( P ) 
n=o 7i ! r (p+?<) 
the double series may be written 
<j>{y) = i r ( " t >ff) 0 F 1 {m + 2 ; y}. (Compare Part X.) 
§ 39. We proceed now to obtain an asymptotic expansion for this integral function 
of y. 
For this purpose we shall anticipate the asymptotic expansions of the hypei- 
geometric series o^i{p ; y], which will be subsequently developed. 
We shall show that asymptotically (§51, III.) 
„F.<P ;!/} = fff y’O Uy} v ~ r A\ p-h i - P ; 
VyJ’ 
wherein {y) > 0 and Vy denotes the positive value of the square root when y is 
real and positive. 
The series 
r , . Plp2,, , Pl(pl+ 1 )(P2)(P2+ 1 ) T 2, 
2 F 0 {pi, p 2 ; x\ = 1 + ^-x + - ^2 
The modulus of the error which results from stopping at any term of the given 
series is less than that of the last term retained when \y | is very large. 
